Conjugate gradient methods for real symmetric systems can be applied
to complex Hermitian systems in a straightforward manner. For
non-Hermitian complex systems we distinguish two cases. In general,
for any coefficient matrix a CGNE method is possible,
that is, a conjugate gradients method on the normal equations
,
or one can split the system into real and complex parts and use
a method such as GMRES on the resulting real nonsymmetric system.
However, in certain practical situations the complex system is
non-Hermitian but symmetric.

Complex symmetric systems can be solved by a classical conjugate
gradient or Lanczos method, that is, with short recurrences, if the
complex inner product is replaced by .
Like the BiConjugate Gradient method, this method is susceptible to
breakdown, that is, it can happen that for .
A look-ahead strategy can remedy this in most
cases (see Freund [100]
and Van der Vorst and Melissen [208]).